
 In [1]} Peterzil and Pillay extracted from [2]} the following. Suppose that \(\mathcal {M}\)  has definable choice (e.g. expands an ordered group). Let \(\mathcal {U}\)  be a monster model and \(\phi (x,b)\)  be a formula in \(\mathcal {L}(U)\)  such that \(\phi (U^{l(x)}, b)\)  is closed and bounded (in the euclidean topology). If the family \(\lbrace  \phi (U^{l(x)},b^{\prime }) : \operatorname{tp}(b^{\prime }/M)=\operatorname{tp}(b/M)\rbrace \)  is consistent, then \(\phi (U^{l(x)},b)\)  has a point in \(M^{l(x)}\) . They derive from this that any definable family of closed and bounded (equivalently definably compact with respect to the euclidean topology) sets that is consistent has a finite transversal [1]}. Our work generalizes these results in a number of ways: we drop the assumption of having definable choice in \(\mathcal {M}\) , and consider any \(M\) -definable topology. We also weaken the intersection assumption (in their work they actually observe that it suffices to have \(k\) -consistency for some \(k\)  in terms of \(l(x)\)  and \(l(b)\) ) without the use of \(\text{VC}\)  or forking literature.

 \((p,q)\) -theorems are closely related to so-called Fractional Helly theorems (see [4]}), which branched from the classical Helly theorem. In its infinite version, this theorem states that any family of closed and bounded convex subsets of \(\mathbb {R}^n\)  that is \((n+1)\) -consistent has nonempty intersection. With an eye towards definably extending Lipschitz maps, Aschenbrenner and Fischer proved ([5]}, Theorem B) a definable version of Helly's Theorem (i.e. for definable families) in definably complete expansions of real closed fields.
Our Theorem REF  and the arguments in Section \(3.2\)  in [5]} allow a generalization of the o-minimal part of Aschenbrenner's and Fischer's definable Helly Theorem, by asking that the sets be definably compact and closed in any definable topology, instead of closed and bounded in the euclidean sense. Moreover, by appropriately adapting Corollary \(2.6\)  in [5]}, one may show that every definable family of convex subsets of \(M^n\)  that is \((n+1)\) -consistent extends to a definable type.


